How thorough? A quick options check and “that will be ’challenging’ and chuffing expensive”?
Note:, I've spent some time putting this response together and as usual Bald Rick has beaten me to it, but here is the way that I could imagine it.
Maybe not as difficult as it superficially looks. I doubt it would tunnel completely under Guildford, - probably starting the bridge under the northbound SW main line, instead of climbing up to clear the Dennis roundabout, take a slight fall and pass under it, resume the climb under the existing alignment (there's so many slips on and off the current 2+2 lane road that there is probably enough room the make it 3+3 lanes without further subterranean land take). The rise to the existing line could be completed before the Guildford Holiday Inn from where the only non trunk access to the A3 is the ridiculous Beechcroft Drive/Manor Way skewed junction.*
I'm not presuming that it wouldn't be an expensive programme, or that it wouldn't be quite disruptive in it's implementation, but the cut and cover activities aren't exactly novel, (M25 Holmesdale Tunnel, A1(M) Hatfield Tunnel etc.), and it would use the gradient down to the railway bridge to avoid the river, and by choosing the correct points to climb that end of the Hog's Back ridge, the gradients would be perfectly acceptable fo a trunk road. The rise from the railway bridge to the Holiday Inn is about 42m in 1.84 km and the current road falls on the western side of the Dennis rounndabout with an almost level stretch past the Midleton Road and Egerton Road junction. Thus there are opportunities to adjust the tunnel gradients as required.
* This junction should really not be there and be replaced with a two lane flyover with southbound A3 access via the Farnham Road slip.